Stochastic averaging technique for SDOF strongly nonlinear systems with delayed feedback fractional-order PD controller

A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom (SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative (PD) controller. The delayed feedback fractional-order PD control force is approximated by an equivalent non-delay feedback control force combining with a quasi-linear elastic force and a quasi-linear damping force. The averaged Itô stochastic differential equation for amplitude of the equivalent nonlinear system is derived by the generalized harmonic functions. The analytical stationary probability density function (PDF) is obtained with solving the reduced Fokker-Planck-Kolmogorov (FPK) equation. Two examples of van der Pol oscillator and Rayleigh-Duffing oscillator are studied to illustrate the application and effectiveness of the proposed method. Numerical results display that the proposed method can yield to the high precision, and the time delay could ruin the control effectiveness, but also even amplifies the response of the system more than that of uncontrolled system. Furthermore, the study finds that the parameters of fractional-order α and time delay may cause the stochastic P-bifurcation. It is indicated that the delayed feedback fractional-order PD controller can offer a potentially effective tool for anti-control of stochastic bifurcation

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